[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.14317/jami.2015.145

AN EFFICIENT SEARCH SPACE IN COUNTING POINTS ON GENUS 3 HYPERELLIPTIC CURVES OVER FINITE FIELDS

Sohn, Gyoyong (Department of Mathematics Education, Deagu National University of Education)

Publication Information

Journal of applied mathematics & informatics / v.33, no.1_2, 2015 , pp. 145-155 More about this Journal

Abstract

In this paper, we study the bounds of the coefficients of the characteristic polynomial of the Frobenius endomorphism of the Jacobian of dimension three over a finite field. We provide explicitly computable bounds for the coefficients of the characteristic polynomial. In addition, we present the counting points algorithm for computing a group of the Jacobian of genus 3 hyperelliptic curves over a finite field with large characteristic. Based on these bounds, we found an efficient search space that was used in the counting points algorithm on genus 3 curves. The algorithm was explained and verified through simple examples.

Keywords

Counting Points; Hyperelliptic Curve; Cryptography;

Citations & Related Records

Reference

1	L. Adleman and M. D. Huang, Counting rational points on curves and abelian varieties over finite fields, in H. Cohen (ed.), ANTS-II, LNCS 1122, Springer-Verlag, (1996) pp. 1-16.
2	I. Blake, G. Seroussi and N. Smart, Elliptic curves in cryptography, London Math. Soc. Lecture Note Series 265 (1999).
3	J. Denef and F. Vercauteren, An Extension of Kedlaya's Algorithm to Hyperelliptic Curves in Characteristic 2, J. Cryptology, 19 (2006), 1-25. DOI
4	N. Elkies, Elliptic and modular curves over finite fields and related computational issues, Computational perspectives on number theory, vol. 7 of AMS/IP Stud. Adv. Math., pp.21-76, Am. Math. Soc. (1998).
5	P. Gaudry and R. Harley, Counting points on hyperelliptic curves over finite fields, ANTS-IV, W. Bosma ed., LNCS 1838 (2000), Springer-Verlag, 297-312.
6	P. Gaudry and E. Schost, A low-memory parallel version of Matsuo, Chao and Tsujii's algorithm, In D. A. Buell, editor, Proceedings of Algorithm Number Theory Sympositum-ANTS VI. volume 3076 of LNCS, pages 208-222, Springer-Verlag, 2004.
7	S. Haloui, The characteristic polynomials of abelian varieties of dimensions 3 over finite fields, J. Number theory, 2011.
8	M. D. Huang and D. Ierardi, Counting points on curves over finite fields, J. Symb. Comp. 25(1) (1998), 1-21. DOI ScienceOn
9	K.S. Kedlaya, Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology, J. Ramanujan Math. Soc. 16 (2001), 323-338.
10	R. Lercier, Algorithmique des courbes elliptiques dans les corps finis. These, Ecole polytechnique, June 1997.
11	Yu. I. Manin, The Hasse-Witt matrix of an algebraic curve, AMS Trans. Ser. 2(45) (1965), 245-264.
12	K. Matsuo, J. Chao, and S. Tsujii, An improved baby step giant step algorithm for point counting of hyperelliptic curves over finite fields, In C. Fiecker and D. Kohel, editors, Proceedings of Algorithm Number Theory Symposium-ANTS V. volume 2369 of LNCS, pages 461-474, Springer-Verlag, 2002.
13	V. Oorschot and P.C., Wiener, M.J., Parallel collusion Search with Cryptanalytic Applications, J. Cryptology 12 (1990), 1-28.
14	J. Pila, Frobenius maps of abelian varieties and finding roots of unity in finite fields, Math. Comp. 55(192) (1990), 745-763. DOI ScienceOn
15	H.-G.Ruck, Abelian surfaces and jacobian varieties over finite fields, Compositio Math. 76(3) (1990), 351-366.
16	T. Satoh, The canonical lift of an ordinary elliptic curve over a finite field and its point counting, J. Ramanujan Math. Soc. 15 (2000), 247-270.
17	R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp. 44 (1985) 483-494.
18	G. Sohn, Pointing algorithm for one-dimensional family of genus 3 nonhyperelliptic curves over finite fields, J. Appl. Math. & Informatics 30 (2012), 101-109.